(untitled) (Communications in Computational Physics, 2025)
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semi-linear partial differential equation with low-regular solutions in two/three dimensions. The method is obtained by combining basis functions from a class of shallow neural networks and the resulting multi-scale analogues, a residual strategy in adaptive methods and the non-overlapping domain decomposition method. Firstly, based on the solution residual, the domain Ω is iteratively decomposed and eventually partitioned into K+1 non-overlapping subdomains, denoted respectively as {Ω_k}_k=0^K, where the exact solution is smooth on subdomain Ω_0 and low-regular on subdomain Ω_k (1≤ k≤ K). Secondly, the low-regular solutions on different subdomains Ω_k (1≤ k≤ K) are approximated by neural networks with different scales, while the smooth solution on subdomain Ω_0 is approximated by the initialized neural network. Thirdly, we determine the undetermined coefficients by solving the linear least squares problems directly or the nonlinear least squares problem via the Gauss-Newton method. The proposed method can be extended to multi-level case naturally. Finally, we use this adaptive method for several peak problems in two/three dimensions to show its high-efficient computational performance.